Gear shifter exhibiting pronounced vibrations at low RPM

[ianwh4]

I am considering the acquisition of my first BMW, namely a 2011 3 Series 320d Exclusive with 55,000 kilometres.
There is no smoke or noise from the engine; it accelerates effectively in all speeds and operates smoothly.
The caveat is that when initiating the gear change, it exhibits strong lateral shaking. A mechanic buddy accompanying me said it may be the flywheel or a mount, while the dealer indicated it might be the gear linkage.
Which scenario is most probable?

[VWDaveBMW]

I would suspect the DMF.

I highly question its connection unless there is much excessive play present.

[ianwh4]

Thank you. I did not see it as very loose, but I had never operated a BMW before, so I was uncertain.

[DreamWeaver]

Any alterations or noises occurring when engaging the clutch.
DMFs often have a little rattling when malfunctioning.
Agreement with VWDaveBMW's statement.

[dave783]

It seems to be the propeller centre bearing, which is an inexpensive component that can be easily replaced; however, the exhaust must be lowered, among other tasks.

[DreamWeaver]

The vehicle is immobile, Dave783; I am inclined to disagree.